Is this the correct way to calculate $MSE$ from $SS_{\text{treatment(s)}}$, $df_{\text{treatment(s)}}$, $F$?

Question

As described in a few of my previous questions (here and there), I am interested in deriving summary statistics from statistics reported in the literature.

I would very much appreciate any advice into the validity or errors found in the following calculations.

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To solve for $MSE$ given $F$, $df_{\text{group}}$, and $SS$.

This is required when a partial anova table is provided.

Given: \begin{equation}\label{eq:f} F = MS_g/MS_e \end{equation}

Where $g$ indicates the group, or treatment. Rearranging this equation gives: $$MS_e=MS_g/F$$

Given

$$MS_x = SS_x/df_x$$

Substitute $SS_g/df_g$ for $MS_g$ in the first equation

$$F=\frac{SS_g/df_g}{MS_e}$$

Then solve for $MS_e$

\begin{equation}\label{eq:mse} MS_e = \frac{SS_g}{df_g\times F} \end{equation}

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Example from table 3 in Starr 2008.

The results are from one (two?) factor ANOVA with repeated measures, with treatment and week as the factors and no replication. "Effects of treatment on individual species were analyzed using Repeated Measures ANOVA in the statistical package Superanova (Abacus Concepts, Berkeley)."

We will calculate MSE from the $SS_{\text{treatment}}$ df_{\text{treatment}}, and $F$-value given in the table; these are $109.58$, $2$, and $0.570$, respectively; $df_{\text{weeks}}$ is given as $10$.

For the 1997 \textit{Eriphorium vaginatum}, the mean $A_{max}$ in table 4 is $13.49$.

Calculate $MS_e$:

$$MS_e = \frac{109.58}{0.57 \times 2} = 96.12$$

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If this is the correct way, then shouldn't $MS_e$ be the same calculated based on factor a or factor b?

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This is only a first draft, but my first attempt at presenting this detailed of a mathematical derivation. Feedback on the writing and presentation would be much appreciated.

Thanks!

Answer 1

Your derivation is perfectly fine for regular (non-repeated meaures) ANOVA. For repeated measures ANOVA the F-statistics does not always equal $MS_g/MS_e$. Assuming weeks is the repeated measure, this formula is correct for the weeks and weeks*treatmeant terms (note that they will give the same MSE), but not for the treatment term. There is another term that is not shown in this table: subjects within treatment, and the MSE you found actually corresponds to that term.

If you are interested in complicated ANOVA tables, I would recommend a rather old book, that has a lot of these formulas worked out:
O. J. Dunn and V.A. Clark: Applied Statistics: Analysis of Variance and Regression, Wiley